NAG C Library Function Document nag_anova_row_col (g04bcc)

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1 1 Purpose NAG C Library Function Document nag_anova_row_col () nag_anova_row_col () computes the analysis of variance for a general row and column design together with the treatment means and standard errors. 2 Specification #include <nag.h> #include <nagg04.h> void nag_anova_row_col (Integer nrep, Integer nrow, Integer ncol, const double y[], Integer nt, const Integer it[], double *gmean, double tmean[], double table[], double c[], Integer tdc, Integer irep[], double rpmean[], double rmean[], double cmean[], double r[], double ef[], double tol, Integer irdf, NagError *fail) 3 Description In a row and column design the experimental material can be characterized by a two-way classification, nominally called rows and columns. Each experimental unit can be considered as being located in a particular row and column. It is assumed that all rows are of the same length and all columns are of the same length. Sets of equal numbers of rows and columns can be grouped together to form replicates, sometimes known as squares or rectangles, as appropriate. If for a replicate, the number of rows, the number of columns and the number of treatments are equal and every treatment occurs once in each row and each column then the design is a Latin square. If this is not the case the treatments will be non-orthogonal to rows and columns. For example in the case of a lattice square each treatment occurs only once in each square. For a row and column design, with t treatments in r rows and c columns and b replicates or squares with n ¼ brc observations, the linear model is: y ijkðlþ ¼ þ i þ j þ k þ l þ e ijk i ¼ 1; 2...;b; j ¼ 1; 2;...;r; k ¼ 1; 2...;c; l ¼ 1; 2;...;t, where i is the effect of the ith replicate, j is the effect of the jth row, k is the effect of the kth column and the ijkðlþ notation indicates that the lth treatment is applied to the unit in row j, column k of replicate i. To compute the analysis of variance for a row and column design the mean is computed and subtracted from the observations to give, y 0 ijkðlþ ¼ y ijkðlþ ^. Since the replicates, rows and columns are orthogonal the estimated effects, ignoring treatment effects, ^ i, ^ j, ^ k, can be computed using the appropriate means of the y 0 ijkðlþ, and the unadjusted sum of squares computed as the appropriate sum of squared totals for the y 0 ijkðlþ divided by number of units per total. The observations adjusted for replicates, rows and columns can then be computed by subtracting the estimated effects from y 0 ijkðlþ to give y00 ijkðlþ. In the case of a Latin square design the treatments are orthogonal to replicates, rows and columns and so the treatment effects, ^ l, can be estimated as the treatment means of the adjusted observations, y 00 ijkðlþ. The treatment sum of squares is computed as the sum of squared treatment totals of the y 00 ijðlþ divided by the number of times each treatment is replicated. Finally the residuals, and hence the residual sum of squares, are given by, r ijðlþ ¼ y 00 ijðlþ ^ l. For a design which is not orthogonal, for example a lattice square or an incomplete Latin square, the treatment effects adjusted for replicates, rows and columns need to be computed. The adjusted treatment effects are found as the solution to the equations: A^ ¼ðR N b N T b =ðrcþ N rn T r =ðbcþ N cn T c =ðbrþþ^ ¼ q.1

2 NAG C Library Manual where q is the vector of the treatment totals of the observations adjusted for replicates, rows and columns, y 00 ijkðlþ ; R is a diagonal matrix with R ll equal to the number of times the lth treatment is replicated, and N b is the t by b incidence matrix, with N l;i equal to the number of times treatment l occurs in replicate i, with N r and N c being similarly defined for rows and columns. The solution to the equations can be written as: ^ ¼ q where, is a generalized inverse of A. The solution is found from the eigenvalue decomposition of A. The residuals are first calculated by subtracting the estimated adjusted treatment effects from the adjusted observations to give r 0 ijðlþ ¼ y00 ijðlþ ^ l. However, since only the unadjusted replicate, row and column effects have been removed and they are not orthogonal to treatments, the replicate, row and column means of the r 0 ijðlþ have to be subtracted to give the correct residuals, r ijðlþ and residual sum of squares. Given the sums of squares, the mean squares are computed as the sums of squares divided by the degrees of freedom. The degrees of freedom for the unadjusted replicates, rows and columns are b 1, r 1 and c 1 respectively and for the Latin square designs the degrees of freedom for the treatments is t 1. In the general case the degrees of freedom for treatments is the rank of the matrix. The F-statistic given by the ratio of the treatment mean square to the residual mean square tests the hypothesis: H 0 : 1 ¼ 2 ¼...¼ t ¼ 0: The standard errors for the difference in treatment effects, or treatment means, for Latin square designs, are given by: p seð^ j ^ j Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2s 2 =ðbtþ where s 2 is the residual mean square. In the general case the variances of the treatment effects are given by: Varð^Þ ¼s 2 from which the appropriate standard errors of the difference between treatment effects or the difference between adjusted means can be calculated. The analysis of a row-column design can be considered as consisting of different strata: the replicate stratum, the rows within replicate and the columns within replicate strata and the units stratum. In the Latin square design all the information on the treatment effects is given at the units stratum. In other designs there may be a loss of information due to the non-orthogonality of treatments and replicates, rows and columns and information on treatments may be available in higher strata. The efficiency of the estimation at the units stratum is given by the (canonical) efficiency factors, these are the non-zero eigenvalues of the matrix, A, divided by the number of replicates in the case of equal replication, or by the mean of the number of replicates in the unequally replicated case, (see John (1987)). If more than one eigenvalue is zero then the design is said to be disconnected and information on some treatment comparisons can only be obtained from higher strata. 4 Parameters 1: nrep Integer Input On entry: the number of replicates, b. Constraint: nrep 1. 2: nrow Integer Input On entry: the number of rows per replicate, r. Constraint: nrow 2. 3: ncol Integer Input On entry: the number of columns per replicate, c. Constraint: ncol 2..2

3 4: y[nrep*nrow*ncol] const double Input On entry: the n ¼ brc observations ordered by columns within rows within replicates. That is y[rcði 1Þþrðj 1Þþk 1] contains the observation from the k column of the jth row of the ith replicate, i ¼ 1; 2;...;b; j ¼ 1; 2...;r; k ¼ 1; 2;...;c. 5: nt Integer Input On entry: the number of treatments. If only replicates, rows and columns are required in the analysis then set nt ¼ 1. Constraint: nt 1. 6: it[dim1] const Integer Input Note: the dimension, dim1, of the array it must be at least nrepnrowncol if nt > 1, and 1 otherwise. On entry: if nt > 1, it½i 1Š indicates which of the nt treatments unit i received, i ¼ 1; 2;...;n.If nt ¼ 1, it is not referenced. Constraint: if nt 2, 1 it½i 1Š nt, for i ¼ 1; 2;...;n. 7: gmean double * Output On exit: the grand mean, ^. 8: tmean[nt] double Output On exit: if nt 2, tmean½l 1Š contains the (adjusted) mean for the lth treatment, ^ þ ^ l, l ¼ 1; 2;...;t, where ^ is the mean of the treatment adjusted observations y ijkðlþ ^ l. Otherwise tmean is not referenced. 9: table[6][5] double Output On exit: the analysis of variance table. Column 1 contains the degrees of freedom, column 2 the sum of squares, and where appropriate, column 3 the mean squares, column 4 the F-statistic and column 5 the significance level of the F-statistic. Row 1 is for replicates, row 2 for rows, row 3 for columns, row 4 for treatments (if nt > 1), row 5 for residual and row 6 for total. Mean squares are computed for all but the total row, F-statistics and significance are computed for treatments, replicates, rows and columns. Any unfilled cells are set to zero. 10: c[nt][tdc] double Output On exit: the upper triangular part of c contains the variance-covariance matrix of the treatment effects, the strictly lower triangular part contains the standard errors of the difference between two treatment effects (means), i.e., c½i 1Š½j 1Š contains the covariance of treatment i and j if j i and the standard error of the difference between treatment i and j if j < i, i ¼ 1; 2;...; t; j ¼ 1; 2;...;t. 11: tdc Integer Input On entry: the second dimension of the array c as declared in the function from which nag_anova_row_col is called. Constraint: tdc nt. 12: irep[nt] Integer Output On exit: if nt > 1, irep½l 1Š contains the treatment replications, R ll, l ¼ 1; 2;...;nt. Otherwise irep is not referenced. 13: rpmean[nrep] double Output On exit: if nrep > 1, rpmean½i 1Š contains the mean for the ith replicate, ^ þ ^ i, i ¼ 1; 2;...;b. Otherwise rpmean is not referenced..3

4 NAG C Library Manual 14: rmean[nrep*nrow] double Output On exit: rmean½j 1Š contains the mean for the jth row, ^ þ ^ i, j ¼ 1; 2;...;r. 15: cmean[nrep*ncol] double Output On exit: cmean½k 1Š contains the mean for the kth column, ^ þ ^ k, k ¼ 1; 2;...;c. 16: r[nrep*nrow*ncol] double Output On exit: r½i 1Š contains the residuals, r i, i ¼ 1; 2;...;n. 17: ef[nt] double Output On exit: if nt 2, the canonical efficiency factors. Otherwise ef is not referenced. 18: tol double Input On entry: the tolerance value used to check for zero eigenvalues of the matrix. Iftol ¼ 0.0 a default value of is used. Constraint: tol : irdf Integer Input On entry: an adjustment to the degrees of freedom for the residual and total. If irdf 1 the degrees of freedom for the total is set to n irdf and the residual degrees of freedom adjusted accordingly. If irdf ¼ 0, the total degrees of freedom for the total is set to n 1, as usual. Constraint: irdf 0. 20: fail NagError * Input/Output The NAG error parameter (see the Essential Introduction). 5 Error Indicators and Warnings NE_INT_ARG_LT On entry, nrep must not be less than 1: nrep ¼ <value>. On entry, nrow must not be less than 2: nrow ¼ <value>. On entry, ncol must not be less than 2: ncol ¼ <value>. On entry, nt must not be less than 1: nt ¼ <value>. On entry, irdf must not be less than 0: irdf ¼ <value>. NE_2_INT_ARG_LT On entry, tdc ¼ <value> while nt ¼ <value>. These parameters must satisfy tdc nt. NE_REAL_ARG_LT On entry, tol must not be less than 0.0: tol ¼ <value>. NE_ARRAY_CONS The contents of array it are not valid. Constraint: if nt 2, 1 it[i] nt, i ¼ 0; 1; 2;...;nrepnrowncol. The contents of array it are not valid. Constraint: some value of it ¼ j for all j ¼ 1,2,...,nt..4

5 NE_ALLOC_FAIL Memory allocation failed. NE_ARRAY_CONSTANT On entry, the elements of the array y are constant. NE_G04BC_ST_ERR A computed standard error is zero due to rounding errors, or the eigenvalue computation failed to converge. Both are unlikely errors. NE_G04BC_REPS The treatments are totally confounded with replicates, rows and columns, so the treatment sum of squares and degrees of freedom are zero. The analysis of variance table is not computed, except for replicate, row, column, total sum of squares and degrees of freedom. NE_G04BC_RESD The residual degrees of freedom or the residual sum of squares are zero, columns 3, 4 and 5 of the analysis of variance table will not be computed and the matrix of standard errors and covariances, c, will not be scaled. NE_G04BC_DISCON The design is disconnected, the standard errors may not be valid. The design may have a nested structure. NE_INTERNAL_ERROR An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance. 6 Further Comments To estimate missing values the Healy and Westmacott procedure or its derivatives may be used (see John and Quenouille (1977)). This is an iterative procedure in which estimates of the missing values are adjusted by subtracting the corresponding values of the residuals. The new estimates are then used in the analysis of variance. This process is repeated until convergence. A suitable initial value may be the grand mean. When using this procedure irdf should be set to the number of missing values plus one to obtain the correct degrees of freedom for the residual sum of squares. For analysis of covariance the residuals are obtained from an analysis of variance of both the response variable and the covariates. The residuals from the response variable are then regressed on the residuals from the covariates using, say, nag_regress_confid_interval (g02cbc) or nag_regsn_mult_linear (g02dac). The results from those routines can be used to test for the significance of the covariates. To test the significance of the treatment effects after fitting the covariate, the residual sum of squares from the regression should be compared with the residual sum of squares obtained from the equivalent regression but using the residuals from fitting replicates, rows and columns only. 6.1 Accuracy The algorithm used in this routine, described in Section 3, achieves greater accuracy than the traditional algorithms based on the subtraction of sums of squares. 6.2 References Cochran W G and Cox G M (1957) Experimental Designs Wiley Davis O L (1978) The Design and Analysis of Industrial Experiments Longman John J A (1987) Cyclic Designs Chapman and Hall.5

6 NAG C Library Manual John J A and Quenouille M H (1977) Experiments: Design and Analysis Griffin Searle S R (1971) Linear Models Wiley 7 See Also nag_regress_confid_interval (g02cbc) nag_regsn_mult_linear (g02dac) 8 Example The data for a 5 5 Latin square is input and the ANOVA and treatment means computed and printed. Since the design is orthogonal only one standard error need be printed 8.1 Program Text /* nag_anova_row_col () Example Program. * * Copyright 2000 Numerical Algorithms Group. * * Mark 6, */ #include <stdio.h> #include <nag.h> #include <nag_stdlib.h> #include <nagg04.h> int main (void) { const char *fmt_99999[]={"%3.0f ", "%10.4f ", "%10.4f ", "%10.4f ", "%8.4f"}; double *c=0, c_b20=1e-5, *cmean=0, *ef=0, gmean, *r=0, *rmean=0, *rpmean=0; double *table=0, *tmean=0, *y=0; Integer c 0=0, i, *irep, *it, j, n, ncol, nrep, nrow, nt; Integer exit_status=0; NagError fail; #define TABLE(I,J) table[((i)-1)*5 + (J)-1] #define C(I,J) c[((i)-1)*nt + (J)-1] INIT_FAIL(fail); Vprintf(" Example Program Results\n"); /* Skip heading in data file */ Vscanf("%*[^\n]"); Vscanf("%ld %ld %ld %ld", &nrep, &nrow, &ncol, &nt); if (!(c = NAG_ALLOC(nt*nt, double))!(cmean = NAG_ALLOC(nrep*ncol, double))!(ef = NAG_ALLOC(nt, double))!(r = NAG_ALLOC(nrep*nrow*ncol, double))!(y = NAG_ALLOC(nrep*nrow*ncol, double))!(rmean = NAG_ALLOC(nrep*nrow, double))!(rpmean = NAG_ALLOC(nrep, double))!(tmean = NAG_ALLOC(nt, double))!(table = NAG_ALLOC(30, double))!(irep = NAG_ALLOC(nt, Integer)).6

7 !(it = NAG_ALLOC(nrep*nrow*ncol, Integer))) { Vprintf("Allocation failure\n"); exit_status = -1; goto END; } n = nrep * nrow * ncol; for (i = 1; i <= n; ++i) Vscanf("%lf", &y[i - 1]); for (i = 1; i <= n; ++i) Vscanf("%ld", &it[i - 1]); (nrep, nrow, ncol, y, nt, it, &gmean, tmean, table, c, nt, irep, rpmean, rmean, cmean, r, ef, c_b20, c 0, &fail); if (fail.code!= NE_NOERROR) { Vprintf("Error from.\n%s\n", fail.message); exit_status = 1; goto END; } Vprintf("\n"); Vprintf("%s\n"," ANOVA TABLE"); Vprintf("\n"); if (nrep > 1) { Vprintf("\n%s", " Reps "); for (j = 1; j <= 5; ++j) Vprintf(fmt_99999[j-1], TABLE(1,j)); } Vprintf("\n%s"," Rows "); for (j = 1; j <= 5; ++j) Vprintf(fmt_99999[j-1], TABLE(2,j)); Vprintf("\n%s", " Columns "); for (j = 1; j <= 5; ++j) Vprintf(fmt_99999[j-1], TABLE(3,j)); Vprintf("\n\n%s", " Treatments "); for (j = 1; j <= 5; ++j) Vprintf(fmt_99999[j-1], TABLE(4,j)); Vprintf("\n%s", " Residual "); for (j = 1; j <= 3; ++j) Vprintf(fmt_99999[j-1], TABLE(5,j)); Vprintf("\n%s", " Total "); for (j = 1; j <= 2; ++j) Vprintf(fmt_99999[j-1], TABLE(6,j)); Vprintf("\n%s\n\n", " Treatment means"); for (i = 1; i <= nt; ++i) Vprintf("%10.4f%s", tmean[i - 1], i%6?"":"\n"); Vprintf("\n\n%s%10.4f\n", " S.E. of difference (orthogonal design) = ", C(2,1)); END: if (c) NAG_FREE(c); if (cmean) NAG_FREE(cmean);.7

8 NAG C Library Manual if (ef) NAG_FREE(ef); if (r) NAG_FREE(r); if (y) NAG_FREE(y); if (rmean) NAG_FREE(rmean); if (rpmean) NAG_FREE(rpmean); if (tmean) NAG_FREE(tmean); if (table) NAG_FREE(table); if (irep) NAG_FREE(irep); if (it) NAG_FREE(it); return exit_status; } 8.2 Program Data Example Program Data Program Results Example Program Results ANOVA TABLE Rows Columns Treatments Residual Total Treatment means S.E. of difference (orthogonal design) =

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